Abstract
This paper is devoted to studying the inhomogeneity of the radiation field resulting from propagation through a multifractal cloud field by relating the orders of singularities and codimensions of both fields. This direct relationship is of fundamental importance for climate studies, whereas the inverse problem is fundamental for remote sensing. We point out similarities between smoothing by scattering and fractional integration, showing they are exactly analogous for certain cases: 1-D medium and plane parallel atmospheres (with a few extra hypotheses). We therefore deduce that there is a limited range of singularities susceptible to exhibiting identical multifractal characteristics before any inversion. The lower bound (γD′) is defined by a first order multifractal phase transition which occurs when the dimension D′ of the fractional integration is insufficient to smooth out the singularities of the cloud field, whereas the upper bound (γs) is defined by a second-order phase transition and corresponds to the limitations induced by the finite size of the samples. These two critical singularities drastically reduce the range of relevant singularities and justify some essentially ad hoc procedures used in multifractal estimation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barker H.W. and J.A. Davies, 1992, Solar radiative fluxes for stochastic, scale invariant Broken Cloud Fields, J. atmos. Sciences, 49, 1115–1126.
Borde R., 1991, Rayonnement dans les nuages multifractals, rapport de stage de D.E.A., Universite de Clermont-Ferrand II, France.
Borde R., E. Gougec, C. Moraillon, D. Schertzer, 1993, Multifractal relationship betweencloud and radiation singularities,exact and asymptotic results,Annales Geophysicae, preprint volume.
Bromsalen G., 1994, Radiative Transfer in lognormal multifractal clouds and analysis of cloud liquid water data, MSc, Mcgill University, Montreal (Quebec), Canada.
Byrne R.N., R.C.J. Somerville, B. Subasilar, 1995, Broken-cloud enhancement of solar radiation absorption, J. Atmos. Sci.
Cahalan R.F., J. H. Joseph, 1989, Fractal statistics of cloud fields, Mon. Wea. Rev., 117, 261–272.
Chandrasekhar S., 1950, Radiative Transfer, Oxford University Press, New York. ( Reprinted by Dover, New York, 1960 ).
Cess R.D., M.H. Zhang, P. Minnis, L. Corsetti, E.G. Dutton, B.W. Forgan, D.P. Garber, W.L. Gates, J.J. Hack, E.F. Harrison, X. Jing, J.T. Kiehl, C.N. Long, J.-J. Morcrette, G.L. Potter, V. Ramanathan, B. Subasilar, C.H. Whitlock, D.F. Young, Y. Zhou, 1995, Absorption of solar radiation by clouds: Observations versus model, Science, 267, 496–499.
Corrsin S., 1951, On the spectrum of isotropic temperature fluctuations in an isotropic turbulence, J.Appl. Phys, 22, 469–473.
Davis A., S. Lovejoy, P.Gabriel, D. Schertzer, G.L. Austin, 1990, Discrete Angle Radiative Transfer Part III: Numerical results on homogeneous and fractal clouds, J. Geophys. Res., 95, 11729–11742.
Davis A., S. Lovejoy, D. Schertzer, 1991, Discrete Angle Radiative Transfer in a multi-fractal medium, Ed. V.V. Varadan, SPIE, 1558, 37–59.
Davis A., S. Lovejoy, D. Schertzer, 1992, Supercomputer simulation of radiative transfer inside multifractal cloud models,I.R.S. 92, A. Arkin et al. Eds., 112–115.
Evans K. F., 1993, A general solution for stochastic radiative transfer, G.R.L., 20, 19, 2075–2078.
Gabriel P., S. Lovejoy, G.L. Austin, D. Schertzer, 1986, Radiative Transfer in extremely variable fractal clouds, 6th conference on atmospheric radiation, AMS, Boston, 230–234.
Gabriel P., S. Lovejoy, D. Schertzer, G.L. Austin, 1988, Multifractal analysis of resolution dependence in satellite imagery, J. Geophys. Res. 15, 1373–1376.
Gabriel P., S. Lovejoy, A.Davis, D. Schertzer, G.L. Austin, 1990, Discrete Angle Radiative Transfer Part II: Renormalization approach to scaling clouds, J. Geophys. Res., 95, 11717–11728.
Kolmogorov A.N., 1941, Local structure of turbulence in an incompressible liquid for very large Reynolds numbers,Proc. Acad. Sci. URSS, Geochem. Sect, 30, 299–303.
Kolmogorov A.N., 1962, A refinement of previous hypothesis concerning the local structure of turbulence in viscous incompressible fluid at high Reynolds numbers. J. Fluid. Mech. 13, 83, 349.
Lavallée D., 1991, Multifractal techniques: Analysis and simulation of turbulent fields, PhD thesis, Mcgill University, Montreal (Quebec), Canada.
Lavallée D., S. Lovejoy, D. Schertzer, P. Ladoy, 1993, Nonlinear variability of landscape topography, multifractal analysis and simulation,in Fractals in geography, eds L. de Cola and N. Lam.
Lévy P., 1925, Calcul des Probabilités, Gauthier Villars, Paris.
Lovejoy S., D. Schertzer, 1989, Fractal clouds with discrete Angle Radiative transfer. I.R.S. 88 Eds. C. Lenoble and J.F. Geylyn, Deepak publishing, 99–102.
Lovejoy S., P. Gabriel, A.Davis, D. Schertzer, G.L. Austin, 1990, Discrete Angle Radiative Transfer Part I: Scaling and similarity, universality and diffusion, J. Geophys. Res., 95, 11699–11715.
Lovejoy S., D. Schertzer, 1990, Multifractals, Universality classes and satellite and radar measurements of cloud and rain fields. J. Geophys. Res. 95, 2021.
Lovejoy S., D. Schertzer, 1991, Multifractal Analysis techniques and the rain and cloud fields from 10–1 to 106m. In Nonlinear Variability in Geophysics: Scaling and Fractals, Kluwer, Schertzer D. and Lovejoy S., 111–144.
Lovejoy S., B. Watson, D. Schertzer, G. Bromsalen, 1995, Scattering in multifractal media. Proc of particle transport in Stochastic Media.
L. Briggs Ed., American Nuclear Society, Portland, Or., April 30-May 4 1995, 750–760.
Lovejoy S. D. Schertzer, P. Silas, 1995, Diffusion on one dimension multifractals. Submitted to Phys. Rev.Lett.
Obukhov A., 1949, Structure of the temperature field in a turbulent flow,IZV. Akad Nauk. SSSR. SER. Geogr. IGeofiz. 13, 55–69.
Obukhov A., 1962, Some specific features of atmospheric turbulence. J. Geophys. Res. 67, 3011.
Parisi G. and Frish U., 1985, A multifractal model of intermittency in turbulence and predictability in geophysical fluid dynamics and climate dynamics, North holland, Ghil M., Benzi R. and Parisi G., pp 84, 88, pp 111, 144.
Pecknold S., S. Lovejoy, D. Schertzer, C. Hooge and J. F. Malouin, 1993, The simulation of universal multifractals. In Cellular Automata: Prospects in astronomy and astrophysics. World Scientific, Perdang J. M. and A. Lejeune.
Ramanathan V., B. Subasilar, G.J. Zhang, W. Conant, R.D. Cess, J.T. Kiehl, H. Grassi, L. Shi, 1995, Warm pool heat budget and shortwave cloud forcing: a missing physics?, Science, 267, 499–503.
Richardson L.F., 1922, Weather prediction by numerical processes, republished by Dover 1965.
Schertzer D., S. Lovejoy, 1987, Physical modelling and analysis of rain and clouds by anisotropic scaling multiplicative processes, J. Geophys. Res. D, 92, 8, 9693–9714.
Schertzer D., S. Lovejoy, 1988, Multifractal simulations and analysis of clouds by multi-plicative processes, Atmos. Res. 21, 337–361.
Schertzer D., S. Lovejoy, 1991, Nonlinear geodynamical variability: multiple singularities, universality and observables. In Nonlinear Variability In Geophysics: scaling and fractals. Kluwer, Schertzer D. and S. Lovejoy, pp 41, 82.
Schertzer D., S. Lovejoy, 1995, The multifractal phase transition route to self-organised criticality. In Physics reports, to appear.
Schertzer D., S. Lovejoy, D. Lavallée, F. Schmitt, 1991, Universal Hard multifractal Turbulence: Theory and Observations. In Nonlinear Dynamics of structures. World Scientific, Sagdeev R.Z. et al., 213.
Schmitt F., D. Lavallée, D. Schertzer, S. Lovejoy, 1992, Empirical determination of universal multifractal exponents in turbulent velocity field, Phys. Rev. Lett., 68, 305.
Schmitt F., D. Schertzer, S. Lovejoy, Y. Brunet, 1993, Estimation of universal multi-fractal indices for atmospheric turbulent velocity fields, Fractals, vol 1, 3, 568–575.
Schmitt F., D. Schertzer, S. Lovejoy, Y. Brunet, 1994, Empirical Study of Multifractal Phase Transitions in Atmospheric turbulence, N.P.G., 1, 95–104.
Schuster A., 1905, Astrophys. J., 21, 1.
Schwartzschild, 1906, Göttinger. Nachrichten, 41.
Silas P. et al., 1993, Single phase diffusion in multifractal porous rock, proceedings Hydrofractals 1993, 1–6.
Silas P., 1994, Diffusion on one dimensional multifractal, MSc, Mcgill University, Montreal (Quebec), Canada.
Somerville R.C.J., C. Gauthier, 1994, Climate-Radiation Feedbacks: The Current State of the Science, in Elements of Change, Eds. S.J. Hassol and P. Norris, Aspen Global Chang Institute.
Tessier Y., S. Lovejoy, Schertzer D., 1993, Universal multifractals: theory and observations for rain and clouds. J. Appl. Meteor. 32, 2, 223–250.
Wilson S., D. Schertzer, S. Lovejoy, 1991, Physically based modelling by multiplicative cascade processes. In Nonlinear Variability in Geophysics: scaling and fractals, Kluwer, Schertzer D. and S. Lovejoy, 185–208.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Naud, C., Schertzer, D., Lovejoy, S. (1997). Radiative Transfer in Multifractal Atmospheres: Fractional Integration, Multifractal Phase Transitions and Inversion Problems. In: Molchanov, S.A., Woyczynski, W.A. (eds) Stochastic Models in Geosystems. The IMA Volumes in Mathematics and its Applications, vol 85. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8500-4_13
Download citation
DOI: https://doi.org/10.1007/978-1-4613-8500-4_13
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-8502-8
Online ISBN: 978-1-4613-8500-4
eBook Packages: Springer Book Archive